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On the isomorphic reduction of an invariant associated with a Lie pseudogroup. (English) Zbl 0602.58057
The paper is devoted to the problem of essential invariants of Lie-Cartan pseudogroups in a slightly generalized setting. In more detail, let \(\pi\) : \(P\to M\) be a fibered manifold, \(u^ 1,...,u^ n\) be functions on P, and \(\omega^ 1,...,\omega^ r\) be 1-forms on P. We consider the pseudogroup \({\mathfrak G}(P)\) on P consisting of all local mappings f that satisfy \(f^*u^ i\equiv u^ i\) and \(f^*\omega^ j\equiv \omega^ j\), and the pseudogroup \({\mathfrak G}(M)\) on M consisting of all mappings g which are projections of the above f, i.e., \(g\circ \pi =\pi \circ f\) with certain \(f\in {\mathfrak G}(P)\). The problem is whether the restriction of the data \(u^ 2,...,u^ n\), \(\omega^ 1,...,\omega^ r\) on a level set \(u^ 1=const\) leads to the relevant restriction of the pseudogroup with the additional property that the projection \({\mathfrak G}(M)\) is the same as before. Assumptions on the structure equations, involutiveness, and specifying of all invariants are not needed; this is the essence of the mentioned generalization according to the common approach.
Reviewer: J.Chrastina
58H05 Pseudogroups and differentiable groupoids